262 XIV INTEGRATION IN LOCALLY COMPACT GROUPS PROBLEMS 1. Let G be a locally compact group containing a compact open subgroup H. For each automorphism u of G, show that mod(^) is a rational number. (Observe that w(H) n H is a subgroup of finite index in both H and #(H).) Show that the set of elements s e G such that AG(,s) = 1 is an open subgroup of G, containing H. 2. Let G be a compact group, /z a Haar measure on G, and u a (continuous) endomor- phism of G such that u(G) is open in G and the kernel Gu = u~l(e) is a finite subgroup of G. (a) Show that there exists a real number h(u) > 0 and an open neighborhood U of e in G such that, for every open set V c U, the set u(V) is open in G and /A(«(V)) = «K)/i(V) (use (14.2.5)). (b) Show that h(u)« Card(G/w(G))/Card(Gu). (Calculate /x(w(G)) in two different ways, using (a) above and (14.4.2).) 3. Let Q* be the set of nonzero rational numbers, endowed with the discrete topology. On the locally compact space G = R xQ* a law of composition is defined by the formula (x, r)(x', r') = (rx' + x, rr'). Show that with respect to this law of composition G is a locally compact group which is locally isomorphic to the additive group R, but not unimodular. 4. (a) Let G be a locally compact group and let % be a continuous homomorphism of G into the multiplicative group C*. Show that if a complex measure v on G is such that Y(S)I/ = %(s)v for all s e G, then v — ax ' ^ where /x is a left Haar measure on G, and a is a complex constant. (b) A complex measure v on G is said to be left quasi-invariant if ^(s)v is equivalent to v (13.15.6) for each s e G. Show that v is left quasi-invariant if and only if v is a measure equivalent (13.15.6) to a left Haar measure ^ on G. (Reduce to the case v ^> 0; use the criterion (b') of (13.15.5), consider the double integral if' where A is compact, / e ^T(G), and / J> 0, and use the theorem of Lebesgue-Fubini.) 5. (a) Let G be a locally compact group and let X, Y be two closed subgroups of G such that X n Y= {e} and such that the set H = XY (i.e., the set of all xy, where x e X and y e Y) contains a neighborhood of e in G. Show that Q is open in G and that the mapping (x, y)\-+xy~l of X x Y onto O, is a homeomorphism (cf. (12.16.12), by considering X X Y as acting on £) by the rule (x, y) • z = xzy~ 1). (b) Let IJLG , fjLX , JW-Y be left Haar measures on G, X, Y, respectively, and let fi be the restriction of JJLG to O. Show that, up to a constant factor, /x, is the image of MX ® (x'1 ' J^-Y) under the homeomorphism (x, y)\-+xy"~l of X x Y onto H, where x denotes the restriction of AG to Y. Deduce that a real-valued function /defined on O is jii-integrable if and only if the function (x9y)^-^f(xy)^G(y)^(y)""1 is (/XX®/XY)- integrable, and that we then have = a where a is a constant independent of/subset P (resp. C) of X is said to be a